Magnetic response of metallic nanoparticles: Geometric and weakly relativistic effects

While the large paramagnetic response measured in certain ensembles of metallic nanoparticles has been assigned to orbital effects of conduction electrons, the spin-orbit coupling has been pointed out as a possible origin of the anomalously large diamagnetic response observed in other cases. Such a relativistic effect, arising from the inhomogeneous electrostatic potential seen by the conduction electrons, might originate from the host ionic lattice, impurities, or the self-consistent confining potential. Here we theoretically investigate the effect of the spin-orbit coupling arising from the confining potential, quantifying its contribution to the zero-field magnetic susceptibility and gauging it against the ones generated by other weakly-relativistic corrections. Two ideal geometries are considered in detail, the sphere and the half-sphere, focusing on the expected increased role of the spin-orbit coupling upon a symmetry reduction, and the application of these results to actual metallic nanoparticles is discussed. The matrix elements of the different weakly-relativistic corrections are obtained and incorporated in a perturbative treatment of the magnetic field, leading to tractable semi-analytical and semiclassical expressions for the case of the sphere, while a numerical treatment becomes necessary for the half-sphere. The correction to the zero-field susceptibility arising from the spin-orbit coupling in a single sphere is quite small, and it is dominated by the weakly-relativistic kinetic energy correction, which in turn remains considerably smaller than the typical values of the nonrelativistic zero-field susceptibility. Moreover, the spin-orbit contribution to the average response for ensembles of nanoparticles with a large size dispersion is shown to vanish. The symmetry reduction in going from the single sphere to the half-sphere does not translate into a significant (...)